Question

If arc BD is two times the arc AC,find

Answer 1

Step-by-step explanation:

To Prove :-

$\sf \dfrac{cot\theta + cosec\theta - 1}{cot\theta - cosec\theta + 1} = \dfrac{1 + cos\theta}{sin\theta}$

Proof:-

$\: \: \: \: \: \: \: \: \: \:\sf: \implies L.H.S$

$\: \: \: \: \: \: \: \pink{ \: \: \::\implies \sf \dfrac{cot\theta + cosec\theta - 1}{cot\theta - cosec\theta + 1}}$

$\: \: \: \: \: \: \: \: \: \::\implies \sf \dfrac{cot\theta + cosec\theta - \big(cosec^2\theta - cot^2\theta\big)}{cot\theta - cosec\theta + 1}$

$\: \: \: \: \: \: \: \: \: \::\implies \sf \dfrac{cot\theta + cosec\theta - \big(cosec\theta + cot\theta\big)\big(cosec\theta - cot\theta\big)}{cot\theta - cosec\theta + 1}$

$\: \: \: \: \: \: \: \: \: \::\implies \sf \dfrac{cot\theta + cosec\theta\Big[1 - \big(cosec\theta - cot\theta\big)\Big]}{cot\theta - cosec\theta + 1}$

$\: \: \: \: \: \: \: \: \: \::\implies \sf \dfrac{cot\theta + cosec\theta\Big[1 - cosec\theta + cot\theta\Big]}{1 - cosec\theta + cot\theta}$

$\: \: \: \: \: \: \: \: \: \::\implies \sf cot\theta + cosec\theta$

$\: \: \: \: \: \: \: \: \: \::\implies \sf \dfrac{cos\theta}{sin\theta} + \dfrac{1}{sin\theta}$

$\: \: \: \: \: \: \: \: \pink{ \: \::\implies \sf \dfrac{1 +cos\theta}{sin\theta}}$

$\: \: \: \: \: \: \: \: \: \:\sf: \implies R.H.S$